The smallest integer that can be added to -2m3 − m + m2 + 1 to make it completely divisible by m + 1 is .

Recall the polynomial remainder theorem: if is a factor of some polynomial , then the remainder upon dividing by is .

Let and . The remainder left from dividing by is

If we subtract 5 from both sides, we'd get a "remainder" of 0, which suggests that we have to add -5 to to make a factor.

This is an arithmetic sequence because there is a common difference between terms, a constant found when subtracting the preceding term from any term in the sequence.  In this case the common difference is 7.

Any arithmetic sequence can be expressed as:

a(n)=a+d(n-1), a=initial value, d=common difference, n=term number

Here we have a=44 and d=7 so

a(n)=44+7(n-1)

a(n)=44+7n-7

a(n)=7n+37, so the 39th term is:

a(37)=7(37)+7

a(37)=266

I am assuming that 44 is the first term, not the 5th term...if 44 was the fifth term let me know and I will edit to reflect that...